direct product, abelian, monomial, 2-elementary
Aliases: C22×C60, SmallGroup(240,185)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C60 |
| C1 — C22×C60 |
| C1 — C22×C60 |
Generators and relations for C22×C60
G = < a,b,c | a2=b2=c60=1, ab=ba, ac=ca, bc=cb >
Subgroups: 108, all normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, C23, C10, C10, C12, C2×C6, C15, C22×C4, C20, C2×C10, C2×C12, C22×C6, C30, C30, C2×C20, C22×C10, C22×C12, C60, C2×C30, C22×C20, C2×C60, C22×C30, C22×C60
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, C23, C10, C12, C2×C6, C15, C22×C4, C20, C2×C10, C2×C12, C22×C6, C30, C2×C20, C22×C10, C22×C12, C60, C2×C30, C22×C20, C2×C60, C22×C30, C22×C60
►Regular action on 240 points
Generators in S240
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 141)(12 142)(13 143)(14 144)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 151)(22 152)(23 153)(24 154)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 175)(46 176)(47 177)(48 178)(49 179)(50 180)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 127)(58 128)(59 129)(60 130)(61 206)(62 207)(63 208)(64 209)(65 210)(66 211)(67 212)(68 213)(69 214)(70 215)(71 216)(72 217)(73 218)(74 219)(75 220)(76 221)(77 222)(78 223)(79 224)(80 225)(81 226)(82 227)(83 228)(84 229)(85 230)(86 231)(87 232)(88 233)(89 234)(90 235)(91 236)(92 237)(93 238)(94 239)(95 240)(96 181)(97 182)(98 183)(99 184)(100 185)(101 186)(102 187)(103 188)(104 189)(105 190)(106 191)(107 192)(108 193)(109 194)(110 195)(111 196)(112 197)(113 198)(114 199)(115 200)(116 201)(117 202)(118 203)(119 204)(120 205)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 61)(17 62)(18 63)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(121 181)(122 182)(123 183)(124 184)(125 185)(126 186)(127 187)(128 188)(129 189)(130 190)(131 191)(132 192)(133 193)(134 194)(135 195)(136 196)(137 197)(138 198)(139 199)(140 200)(141 201)(142 202)(143 203)(144 204)(145 205)(146 206)(147 207)(148 208)(149 209)(150 210)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
G:=sub<Sym(240)| (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,127)(58,128)(59,129)(60,130)(61,206)(62,207)(63,208)(64,209)(65,210)(66,211)(67,212)(68,213)(69,214)(70,215)(71,216)(72,217)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,225)(81,226)(82,227)(83,228)(84,229)(85,230)(86,231)(87,232)(88,233)(89,234)(90,235)(91,236)(92,237)(93,238)(94,239)(95,240)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,190)(106,191)(107,192)(108,193)(109,194)(110,195)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,203)(119,204)(120,205), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)>;
G:=Group( (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,127)(58,128)(59,129)(60,130)(61,206)(62,207)(63,208)(64,209)(65,210)(66,211)(67,212)(68,213)(69,214)(70,215)(71,216)(72,217)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,225)(81,226)(82,227)(83,228)(84,229)(85,230)(86,231)(87,232)(88,233)(89,234)(90,235)(91,236)(92,237)(93,238)(94,239)(95,240)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,190)(106,191)(107,192)(108,193)(109,194)(110,195)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,203)(119,204)(120,205), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,61)(17,62)(18,63)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,82)(38,83)(39,84)(40,85)(41,86)(42,87)(43,88)(44,89)(45,90)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240) );
G=PermutationGroup([[(1,131),(2,132),(3,133),(4,134),(5,135),(6,136),(7,137),(8,138),(9,139),(10,140),(11,141),(12,142),(13,143),(14,144),(15,145),(16,146),(17,147),(18,148),(19,149),(20,150),(21,151),(22,152),(23,153),(24,154),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,161),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,169),(40,170),(41,171),(42,172),(43,173),(44,174),(45,175),(46,176),(47,177),(48,178),(49,179),(50,180),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,127),(58,128),(59,129),(60,130),(61,206),(62,207),(63,208),(64,209),(65,210),(66,211),(67,212),(68,213),(69,214),(70,215),(71,216),(72,217),(73,218),(74,219),(75,220),(76,221),(77,222),(78,223),(79,224),(80,225),(81,226),(82,227),(83,228),(84,229),(85,230),(86,231),(87,232),(88,233),(89,234),(90,235),(91,236),(92,237),(93,238),(94,239),(95,240),(96,181),(97,182),(98,183),(99,184),(100,185),(101,186),(102,187),(103,188),(104,189),(105,190),(106,191),(107,192),(108,193),(109,194),(110,195),(111,196),(112,197),(113,198),(114,199),(115,200),(116,201),(117,202),(118,203),(119,204),(120,205)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,61),(17,62),(18,63),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,81),(37,82),(38,83),(39,84),(40,85),(41,86),(42,87),(43,88),(44,89),(45,90),(46,91),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97),(53,98),(54,99),(55,100),(56,101),(57,102),(58,103),(59,104),(60,105),(121,181),(122,182),(123,183),(124,184),(125,185),(126,186),(127,187),(128,188),(129,189),(130,190),(131,191),(132,192),(133,193),(134,194),(135,195),(136,196),(137,197),(138,198),(139,199),(140,200),(141,201),(142,202),(143,203),(144,204),(145,205),(146,206),(147,207),(148,208),(149,209),(150,210),(151,211),(152,212),(153,213),(154,214),(155,215),(156,216),(157,217),(158,218),(159,219),(160,220),(161,221),(162,222),(163,223),(164,224),(165,225),(166,226),(167,227),(168,228),(169,229),(170,230),(171,231),(172,232),(173,233),(174,234),(175,235),(176,236),(177,237),(178,238),(179,239),(180,240)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)]])
C22×C60 is a maximal subgroup of
C60.212D4 C30.29C42 C60.205D4 C23.26D30 C23.28D30 C60⋊29D4
240 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 6A | ··· | 6N | 10A | ··· | 10AB | 12A | ··· | 12P | 15A | ··· | 15H | 20A | ··· | 20AF | 30A | ··· | 30BD | 60A | ··· | 60BL |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
240 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 |
| kernel | C22×C60 | C2×C60 | C22×C30 | C22×C20 | C2×C30 | C22×C12 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C2×C10 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 2 | 8 | 4 | 12 | 2 | 24 | 4 | 16 | 8 | 32 | 48 | 8 | 64 |
Matrix representation of C22×C60 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 60 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 1 |
| 35 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 6 |
G:=sub<GL(3,GF(61))| [60,0,0,0,1,0,0,0,1],[60,0,0,0,60,0,0,0,1],[35,0,0,0,16,0,0,0,6] >;
C22×C60 in GAP, Magma, Sage, TeX
C_2^2\times C_{60} % in TeX
G:=Group("C2^2xC60"); // GroupNames label
G:=SmallGroup(240,185);
// by ID
G=gap.SmallGroup(240,185);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-2,720]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^60=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations